Page 196 - DMTH503_TOPOLOGY

Topology Richa Nandra, Lovely Professional University

Notes Unit 22: The Tychonoff Theorem

CONTENTS
Objectives
Introduction

22.1 Finite Intersection Property
22.2 Summary
22.3 Keywords

22.4 Review Questions
22.5 Further Readings

Objectives

After studying this unit, you will be able to:
 Define finite intersection property;

 Solve the problems on finite intersection;
 Understand the proof of Tychonoff’s theorem.

Introduction

Like the Urysohn Lemma, the Tychonoff theorem is what we call a “deep” theorem. Its proof
involves not one but several original ideas; it is anything but straightforward. We shall prove
the Tychonoff theorem, to the effect that arbitrary products of compact spaces are compact. The
proof makes use of Zorn’s lemma. The Tychonoff theorem is of great usefulness to analysts we
apply it to construct the Stone-Cech compactification of a completely regular space and in
proving the general version of Ascoli’s theorem.

22.1 Finite Intersection Property

Let X be a set and f a family of subsets of X. Then  is said to have the finite intersection property
if for any finite number F , F , ...., F of members of .
1 2 n
F F  .... F 
1 2 n
Proposition: Let (X, ) be a topological space. Then (X, ) is compact if any only if every family
of closed subsets of X with the finite intersection property satisfies  F .
F 
Proof: Assume that every family of closed subsets of X with the finite intersection property
satisfies  F . Let be any open covering of X. Put  equal to the family of complements
F 
of members of . So each F is closed in (X, ). As  is an open covering of X,  F = . By our
F 
assumption, then,  does not have the finite intersection property. So for some F , F ,...F in ,
1 2 n
F  F ....,  F = .
1 2 n
Thus   ...  = X, where
1 2 n
 = X\F, i = 1,...., n.
i i
So  has a finite subcovering. Hence, (X, T) is compact.
The converse statement is proved similarly.

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